General Formulas and Concepts:
Order of Operations: BPEMDAS
- Multiplication Property of Equality
- Division Property of Equality
- Addition Property of Equality
- Subtract Property of Equality
Volume of a Cone:
Diameter: d = 2r
Differentiating with respect to time
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Step 1: Define
Step 2: Rewrite Volume Formula
We need to rewrite the cone volume formula in terms of height h only.
Base b = diameter d of the circular base
- Define: b = d
- Substitute: b = 2r
We are given that the base of the cone is the same as the height.
- Define: b = 2r
- Substitute: h = 2r
Now solve for height.
- Divide 2 on both sides: h/2 = r
- Rewrite expression: r = h/2
Now find new volume formula.
- Define [VC]:
We now have the same volume formula in terms of height h only.
Step 3: Differentiate
- Basic Power Rule:
Step 4: Solve for height rate
- Isolate h rate:
Here this tells us that the rate at which the height is moving at a rate of 0.101859 feet per minute.